Survey on Generalization Theory for Graph Neural Networks

Message-passing graph neural networks (MPNNs) have emerged as the leading approach for machine learning on graphs, attracting significant attention in recent years. While a large set of works explored the expressivity of MPNNs, i.e., their ability to separate graphs and approximate functions over them, comparatively less attention has been directed toward investigating their generalization abilities, i.e., making meaningful predictions beyond the training data. Here, we systematically review the existing literature on the generalization abilities of MPNNs. We analyze the strengths and limitations of various studies in these domains, providing insights into their methodologies and findings. Furthermore, we identify potential avenues for future research, aiming to deepen our understanding of the generalization abilities of MPNNs.

Higher-Order Graphon Neural Networks: Approximation and Cut Distance

Graph limit models, like graphons for limits of dense graphs, have recently been used to study size transferability of graph neural networks (GNNs). While most literature focuses on message passing GNNs (MPNNs), in this work we attend to the more powerful higher-order GNNs. First, we extend the $k$-WL test for graphons (B\"oker, 2023) to the graphon-signal space and introduce signal-weighted homomorphism densities as a key tool. As an exemplary focus, we generalize Invariant Graph Networks (IGNs) to graphons, proposing Invariant Graphon Networks (IWNs) defined via a subset of the IGN basis corresponding to bounded linear operators. Even with this restricted basis, we show that IWNs of order $k$ are at least as powerful as the $k$-WL test, and we establish universal approximation results for graphon-signals in $L^p$ distances. This significantly extends the prior work of Cai & Wang (2022), showing that IWNs--a subset of their IGN-small--retain effectively the same expressivity as the full IGN basis in the limit. In contrast to their approach, our blueprint of IWNs also aligns better with the geometry of graphon space, for example facilitating comparability to MPNNs. We highlight that, while typical higher-order GNNs are discontinuous w.r.t. cut distance--which causes their lack of convergence and is inherently tied to the definition of $k$-WL--their transferability remains comparable to MPNNs.

Beyond Matryoshka: Revisiting Sparse Coding for Adaptive Representation

Many large-scale systems rely on high-quality deep representations (embeddings) to facilitate tasks like retrieval, search, and generative modeling. Matryoshka Representation Learning (MRL) recently emerged as a solution for adaptive embedding lengths, but it requires full model retraining and suffers from noticeable performance degradations at short lengths. In this paper, we show that sparse coding offers a compelling alternative for achieving adaptive representation with minimal overhead and higher fidelity. We propose Contrastive Sparse Representation (CSR), a method that sparsifies pre-trained embeddings into a high-dimensional but selectively activated feature space. By leveraging lightweight autoencoding and task-aware contrastive objectives, CSR preserves semantic quality while allowing flexible, cost-effective inference at different sparsity levels. Extensive experiments on image, text, and multimodal benchmarks demonstrate that CSR consistently outperforms MRL in terms of both accuracy and retrieval speed-often by large margins-while also cutting training time to a fraction of that required by MRL. Our results establish sparse coding as a powerful paradigm for adaptive representation learning in real-world applications where efficiency and fidelity are both paramount. Code is available at https://github.com/neilwen987/CSR_Adaptive_Rep

Learning with Exact Invariances in Polynomial Time

We study the statistical-computational trade-offs for learning with exact invariances (or symmetries) using kernel regression. Traditional methods, such as data augmentation, group averaging, canonicalization, and frame-averaging, either fail to provide a polynomial-time solution or are not applicable in the kernel setting. However, with oracle access to the geometric properties of the input space, we propose a polynomial-time algorithm that learns a classifier with \emph{exact} invariances. Moreover, our approach achieves the same excess population risk (or generalization error) as the original kernel regression problem. To the best of our knowledge, this is the first polynomial-time algorithm to achieve exact (not approximate) invariances in this context. Our proof leverages tools from differential geometry, spectral theory, and optimization. A key result in our development is a new reformulation of the problem of learning under invariances as optimizing an infinite number of linearly constrained convex quadratic programs, which may be of independent interest.

Subsampling Graphs with GNN Performance Guarantees

How can we subsample graph data so that a graph neural network (GNN) trained on the subsample achieves performance comparable to training on the full dataset? This question is of fundamental interest, as smaller datasets reduce labeling costs, storage requirements, and computational resources needed for training. Selecting an effective subset is challenging: a poorly chosen subsample can severely degrade model performance, and empirically testing multiple subsets for quality obviates the benefits of subsampling. Therefore, it is critical that subsampling comes with guarantees on model performance. In this work, we introduce new subsampling methods for graph datasets that leverage the Tree Mover's Distance to reduce both the number of graphs and the size of individual graphs. To our knowledge, our approach is the first that is supported by rigorous theoretical guarantees: we prove that training a GNN on the subsampled data results in a bounded increase in loss compared to training on the full dataset. Unlike existing methods, our approach is both model-agnostic, requiring minimal assumptions about the GNN architecture, and label-agnostic, eliminating the need to label the full training set. This enables subsampling early in the model development pipeline (before data annotation, model selection, and hyperparameter tuning) reducing costs and resources needed for storage, labeling, and training. We validate our theoretical results with experiments showing that our approach outperforms existing subsampling methods across multiple datasets.

When More is Less: Understanding Chain-of-Thought Length in LLMs

Chain-of-thought (CoT) reasoning enhances the multi-step reasoning capabilities of large language models (LLMs) by breaking complex tasks into smaller, manageable sub-tasks. Researchers have been exploring ways to guide models to generate more complex CoT processes to improve the reasoning ability of LLMs, such as long CoT and the test-time scaling law. However, for most models and tasks, does an increase in CoT length consistently lead to improved reasoning accuracy? In this paper, we observe a nuanced relationship: as the number of reasoning steps increases, performance initially improves but eventually decreases. To understand this phenomenon, we provide a piece of evidence that longer reasoning processes are increasingly susceptible to noise. We theoretically prove the existence of an optimal CoT length and derive a scaling law for this optimal length based on model capability and task difficulty. Inspired by our theory, we conduct experiments on both synthetic and real world datasets and propose Length-filtered Vote to alleviate the effects of excessively long or short CoTs. Our findings highlight the critical need to calibrate CoT length to align with model capabilities and task demands, offering a principled framework for optimizing multi-step reasoning in LLMs.

On the Emergence of Position Bias in Transformers

Recent studies have revealed various manifestations of position bias in transformer architectures, from the "lost-in-the-middle" phenomenon to attention sinks, yet a comprehensive theoretical understanding of how attention masks and positional encodings shape these biases remains elusive. This paper introduces a novel graph-theoretic framework to analyze position bias in multi-layer attention. Modeling attention masks as directed graphs, we quantify how tokens interact with contextual information based on their sequential positions. We uncover two key insights: First, causal masking inherently biases attention toward earlier positions, as tokens in deeper layers attend to increasingly more contextualized representations of earlier tokens. Second, we characterize the competing effects of the causal mask and relative positional encodings, such as the decay mask and rotary positional encoding (RoPE): while both mechanisms introduce distance-based decay within individual attention maps, their aggregate effect across multiple attention layers -- coupled with the causal mask -- leads to a trade-off between the long-term decay effects and the cumulative importance of early sequence positions. Through controlled numerical experiments, we not only validate our theoretical findings but also reproduce position biases observed in real-world LLMs. Our framework offers a principled foundation for understanding positional biases in transformers, shedding light on the complex interplay of attention mechanism components and guiding more informed architectural design.

Understanding the Role of Equivariance in Self-supervised Learning

Contrastive learning has been a leading paradigm for self-supervised learning, but it is widely observed that it comes at the price of sacrificing useful features (\eg colors) by being invariant to data augmentations. Given this limitation, there has been a surge of interest in equivariant self-supervised learning (E-SSL) that learns features to be augmentation-aware. However, even for the simplest rotation prediction method, there is a lack of rigorous understanding of why, when, and how E-SSL learns useful features for downstream tasks. To bridge this gap between practice and theory, we establish an information-theoretic perspective to understand the generalization ability of E-SSL. In particular, we identify a critical explaining-away effect in E-SSL that creates a synergy between the equivariant and classification tasks. This synergy effect encourages models to extract class-relevant features to improve its equivariant prediction, which, in turn, benefits downstream tasks requiring semantic features. Based on this perspective, we theoretically analyze the influence of data transformations and reveal several principles for practical designs of E-SSL. Our theory not only aligns well with existing E-SSL methods but also sheds light on new directions by exploring the benefits of model equivariance. We believe that a theoretically grounded understanding on the role of equivariance would inspire more principled and advanced designs in this field. Code is available at https://github.com/kaotty/Understanding-ESSL.

Generalization, Expressivity, and Universality of Graph Neural Networks on Attributed Graphs

We analyze the universality and generalization of graph neural networks (GNNs) on attributed graphs, i.e., with node attributes. To this end, we propose pseudometrics over the space of all attributed graphs that describe the fine-grained expressivity of GNNs. Namely, GNNs are both Lipschitz continuous with respect to our pseudometrics and can separate attributed graphs that are distant in the metric. Moreover, we prove that the space of all attributed graphs is relatively compact with respect to our metrics. Based on these properties, we prove a universal approximation theorem for GNNs and generalization bounds for GNNs on any data distribution of attributed graphs. The proposed metrics compute the similarity between the structures of attributed graphs via a hierarchical optimal transport between computation trees. Our work extends and unites previous approaches which either derived theory only for graphs with no attributes, derived compact metrics under which GNNs are continuous but without separation power, or derived metrics under which GNNs are continuous and separate points but the space of graphs is not relatively compact, which prevents universal approximation and generalization analysis.

An Information Criterion for Controlled Disentanglement of Multimodal Data

Multimodal representation learning seeks to relate and decompose information inherent in multiple modalities. By disentangling modality-specific information from information that is shared across modalities, we can improve interpretability and robustness and enable downstream tasks such as the generation of counterfactual outcomes. Separating the two types of information is challenging since they are often deeply entangled in many real-world applications. We propose Disentangled Self-Supervised Learning (DisentangledSSL), a novel self-supervised approach for learning disentangled representations. We present a comprehensive analysis of the optimality of each disentangled representation, particularly focusing on the scenario not covered in prior work where the so-called Minimum Necessary Information (MNI) point is not attainable. We demonstrate that DisentangledSSL successfully learns shared and modality-specific features on multiple synthetic and real-world datasets and consistently outperforms baselines on various downstream tasks, including prediction tasks for vision-language data, as well as molecule-phenotype retrieval tasks for biological data.